Special relativistic equilibrium statistics
1. Introduction
After establish of the special theory of relativity by Albert Einstein in 1905, the special
relativistic mechanics and electrodynamics have been completely established. The special
relativistic thermodynamics launched by Planck and Einstein in 1907 has encountered
difficulties especially since 1960s. From our point of view, the key difficulty is that the
motion between any two inertial reference frames in the special theory of relativity is a global
and ordered motion, but the motion which the thermodynamics deals with is disordered
motion. The global and ordered motion gives effects to the disordered motion. For example,
the contraction of length changes the volume, and the pressure does work. The disordered
motion also effects the global and ordered motion. For example, the change of the inner
energy of the system causes the total energy and total momentum of the system to change. We
introduce the total energy E, enthalpy H ( in 1977, Pathria pointed its importance in
relativistic thermodynamics) and inner energy , and establish their relation among them and
then on this basis establish the special relativistic thermodynamics and statistics. The total
energy, including the global and ordered energy, pressure energy and disordered energy, of the
system is the forth component of the total momentum of the system which obey the Lorentz
transformation as the same as that in the special theory of relativity. Enthalpy includes
pressure energy but excludes the global and ordered energy of the system. Inner energy,
excluding both of the global and ordered energy and pressure energy, obeys the first law of
special relativistic thermodynamics. The special relativistic equilibrium thermodynamics
should consider the heat flux and this is an important defference from non-relativistic
thermodynamics, because the motion of the system as a whole need inevitably to introduce
the heat flux. In this case, we have to deal with the thermodynamical equilibrium under the
condition of dynamical equilibrium. That is, either the temperature or the presure does not
need to be homogenious. This inhomogenious temperature distribution corresponds to the
process of quisi-static equilibrium, as the time elapes, the system tends to thermal equilibrium.
The special relativistic thermodynamics and statistics we established in such a way in the
context of present physics is itself consistent and Lorentz covariant. The non-special
relativistic thermodynamics and statistics (also quantum statistics) are all included as a
non-relativistic limit. Some important and new expectations have been given and waiting for
experimental tests. The difficulties encountered have been overcome and found the reasons
and the ways to solve.
2. Phase space
Under the same macroscopic conditions, a macroscopic state of thermodynamics is
consist of a large number of microscopic state. The Hamiltonian of the system correspondent
to the total energy E of the system, which includes the disordered energy-enthalpy H and
ordered energy of the global motion of the system as a whole.
In a finite macroscopic volume V, there are N identical particles with interaction. Each
particle has its rest mass m0 , 4-coordinates q  , 4-velocity u  and 4-momentum
p  m0 u ,m0   0 / c 2 ,
where  0 is the
(4.7.1)
energy of the particle including pressure energy per particle. This is an
important difference between relativistic mechanics and relativistic thermodynamics.
The coordinates and momenta of N particles form a 8N-dimension phase space. The
interaction between particles gives restrains to the system. The freedom s of the system equals
to 8N minus the number of restrains. A microscopic state of the thermodynamically
macroscopic state of the system is determined by a point of the phase space.
The Hamiltonian of the system is H, eigen-state vector is B , the eigenvalue of H is the
total energy E (the pressure energy and moving energy of the system as a whole are included)
of the system and the eigen-equation is as following
H B ＝E B ,
(4.7.2)
where H＝H(q, p) is an operator in quantum mechanics.
The evolution in time of the system obeys the Lorentz covariant Hamiltonian equations
d q  /d  ＝H/ p ;
d p  /dτ＝－H/ q  , (4.7.3)
where  is the proper time.
3. Distribution function and probability density function
The volume element of the phase space is
dqdp＝d q1 d q 2 ...d q s d p1 d p 2 ...d p s . (4.7.4)
The states of the system under a group of macroscopic conditions are distributed in an area of
the phase space. Suppose the momentum of the i-th particle is in the between pi and

pi

 dpi
, its probability in the surface element d i of the 3-dimension hypersurface, which
its world line intersects with, is (Liboff 1990)
ui d i d 4 pi ，
(4.7.5)
and the condition of normalization is
 u  d i d

i
4
pi  constant,
(4.7.6)
where the direction of the tangent vector of the world line of each particle is that of its
4-velocity, ρ is the distribution function or distribution density, which in general is the
function of 4-coordinates and 4-momentum of all particles
 
   ( q , p ),
and satisfies the following conditions:
  0；
(4.7.7)
(4.7.8)
  0，当
p   .
(4.7.9)
Because of the Hamiltonian equations, it can be proved that there is the Lorentz covariant
Liouville’s theorem for each particle as
d / d ＝0.
(4.7.10)
If we consider the system consisting of N particles in 3-coordinate and 3-momentum
phase space,
f ( p)d 3 p =d 3 p  f ( x , p)d 3 x
V
(4.7.11)
expresses the particle number in the momentum volume d 3 p with the momentum nearby p
and the space volume, and it should be Lorentz covariant (the distribution function f should be
normalized to the total particle number N). By the same reason,
f ( x )d 3 x =d 3 x  f ( x , p)d 3 p
V
(4.7.12)
should be Lorentz covariant too.
In the 4-momentum space, for the hypersurface
p  p   m2 c 2 ,
its surface element is
d 3 p= dpx dp y dpz  d 0 .
(4.7.13)
The general law for the Lorentz covariance (Landau and Lifshitz 1987) is that the ratio of
parallel components of two 4-vectors is Lorentz invariant. Therefore
d 0 d 3 p d 3 p


p0
p0 E / c
(4.7.14)
should be Lorentz covariant.
By the similar discussion, or by using the Lorentz transformation of energy and that of
the 3-volume, we arrive the results: distribution function f ( x , p) and volume element
d 3 xd 3 p of the phase space are Lorentz invariant respectively.
Because the classical statistics should be the limit of quantum statistics under the
condition
nh 3
 1 ,
(2 mk B T ) 3/ 2
(4.7.15)
where h is Planck constant, we should use dimensionless probability density function instead
of the distribution function f, and the condition of normalization becomes
 (q, p, t )d ,
d  ( N ! h 3 N ) 1 dqdp ,(4.7.16)
where q expresses the 3-coordinates of all particles of the system, p expresses 3-momentum
of all particles of the system, and the integral is for whole phase space consisting of
3-coordinates and 3-momentum of all particles of the system.
The volume elements d 4 x and d 4 p are invariant due to the orthogonal property of
the Lorentz transformation.
3. Ensemble theory of relativistic statistics
In quantum mechanics, the observed value of the observable of a operator A of the
system is
 c 
2
n
A 
n
n
 cn
, A n 
,
2
(4.1.2)
n
where we have used the hypothesis of “equal-probability”(the system can be in a arbitrary
microscopic state with equal probability) and hypothesis of “random phase”( the linear
combination of stationary wave functions is incoherent) for all cn .
The postulate of random phases implies that the state of a system in equilibrium may be
regarded as an incoherent superposition of eigenstates. It is possible to think of the system as
one member of an infinite collection of systems, each of which is in an eigenstate whose wave
function is  n . Since these systems do not interfere with one another, it is possible to form a
mental picture of each system one at a time. We call this mental picture an ensemble of
systems (Huang 1963).
In the relativistic statistics, any system, which is in the macroscopic thermodynamically
equilibrium with definite temperature T, pressure P and chemical potential , can be described
by general T-P ensemble (Xia 1997). The probability density function is
  1 exp(– H – N) ,
Z
(4.7.32)
where  ＝ 1/ k B T, k B is the Boltzmann’s constant, T is the temperature, H is the
thermodynamical enthalpy, N is the particle number,  is the chemical potential and Z is the
partition function:
Z   H , N e x p(H  N ) .
(4.7.33)
According to the standard way of statistics, for the most probable distribution, the average
particle number in each
ni 
i
e x p(H  N )  g
,
(4.7.34)
where g＝0, 1 and －1 corresponding to Boltzmann, Fermi and Bose statistics respectively,
 i is the degeneracy of the i-th microscopic state.
From the Lorentz transformations of thermal enthalpy H, temperature T and chemical
potential , it is obvious that the probability density function expressed by (4.7.32) is Lorentz
invariant.
From (4.7.32), in the cases of N fixed, we get back to T-P ensemble and (4.7.32)
becomes

 ( H ) e x p(H )
,
  ( H ) e x p(H )
(4.7.35)
H
where  ( H ) is the degeneracy of the microscopic states with the same enthalpy H, while
 ( H ) will be different for different H.
In the cases of T, V and  fixed, but inner energy  and particle number N variable, we
get back to grand canonical ensemble and (4.7.32) becomes

 ( N , ) exp(  N  )
.
  ( N , ) exp(N  )
(4.7.36)
N ,
For the both cases of T-P ensemble with V fixed or the grand canonical ensemble with N
fixed, we all get back to canonical ensemble(Chen 1997). Both of (4.7.35) and (4.7.36)
becomes

 (  ) exp(   )
.
  ( ) exp(  )
(4.7.37)

Furthermore, when inner energy  fixed, we get back to the microscopic canonical
ensemble and (4.7.37) becomes

1

.
(4.7.38)
This is the case that the postulate of equal a priori probability holds.
Because that , , H and T all have the same Lorentz transformation, all probability
density functions for different ensembles are Lorentz invariant.
4. Formula of relativistic thermodynamics
From the Lorentz invariant probability density function (4.7.32), according to the
standard way of thermodynamics and statistics, we can obtain all results of relativistic
thermodynamics.
We start from the generalized T-P ensemble to obtain formula of relativistic
thermodynamics.
The average particle number <N> of the system is the average value of particle number
N under the conditions of fixed H, T and  over all possible microscopic states. From (4.7.17),
we have
 N 
1
Z

 N exp(– H – N) = –  ln Z , (4.7.45)
H,N
where ＝ and Z is given by (4.7.33).
The average thermodynamical enthalpy <H> of the system is the average value of
thermodynamical enthalpy H under the conditions of fixed N, T and  over all possible
microscopic states. From (4.7.17), we have
 H  1  Hexp(– H – N)= –  ln Z . .(4.7.46)
Z H,N

The average volume <V> of the system is the average value of H/P. From (4.7.17), we
have
 V 
1 
ln Z .
 P
(4.7.47)
The first equation of (4.3.20) becomes
dH  TdS  VdP  dN .
(4.7.48)
By using (4.7.45), (4.7.46) and (4.7.47), we consider
 (dH  VdP  dN )   d(
lnZ lnZ
lnZ
)–
dP  d(
),

P

and the total derivatives of lnZ with respect to T, P and  (that is as the function of ,  and
P)
d l nZ =
lnZ
 ln Z
 ln Z
d 
d 
dP ,


P
we obtain
 (dH – VdP  dN )  d(lnZ – 
lnZ
lnZ

) .(4.7.49)


Comparing (4.7.48) and (4.7.49), we have
 1 ,
 

,
k BT
S  k B (ln Z   ln Z    ln Z ).
k BT

(4.7.50)

Up to now, we have obtained all thermodynamical functions of the system in relativistic
thermodynamics.
Before the end of the paper, we would like to give some brief comments.
After establish of the special theory of relativity by Albert Einstein in 1905, The special
relativistic mechanics and electrodynamics have been completely established. The special
relativistic thermodynamics launched by Planck and Einstein in 1907 has encounter
difficulties especially since 1960s. From our point of view, the key difficulty is that the
motion between any two inertial reference frames in the special theory of relativity is a global
and ordered motion, but the motion which the thermodynamics deals with is disordered
motion. The global and ordered motion gives effects to the disordered motion. For example,
the contraction of length changes the volume, and the pressure does work. The disordered
motion also effects the global and ordered motion. For example, the change of the inner
energy of the system causes the total energy and total momentum of the system to change. We
introduce the total energy E, enthalpy H and inner energy , and establish their relation
among them and then on this basis establish the special relativistic thermodynamics and
statistics. The total energy, including the global and ordered energy, pressure energy and
disordered energy, of the system is the forth component of the total momentum of the system
which obey the Lorentz transformation as the same as that in the special theory of relativity.
Enthalpy includes pressure energy but excludes the global and ordered energy of the system.
Inner energy, excluding both of the global and ordered energy and pressure energy, obeys the
first law of special relativistic thermodynamics. The special relativistic equilibrium
thermodynamics should consider the heat flux and this is an important defference from
non-relativistic thermodynamics, because the motion of the system as a whole need inevitably
to introduce the heat flux. In this case, we have to deal with the thermodynamical equilibrium
under the condition of dynamical equilibrium. That is, either the temperature or the presure
does not need to be homogenious. This inhomogenious temperature distribution corresponds
to the process of quisi-static equilibrium, as the time elapes, the system tends to thermal
equilibrium. The special relativistic thermodynamics and statistics we established in such a
way in the context of present physics is itself consistent and Lorentz covariant. The
non-special relativistic thermodynamics and statistics (also quantum statistics) are all
included as a non-relativistic limit. Some important and new expectations have been given
and waiting for experimental tests. The difficulties encountered have been overcome and
found the reasons and the ways to solve.
As a conclusion, we come to the statement that a complete special relativistic
thermodynamics and statistics has been established. On this basis, a general relativistic
thermodynamics and statistics in gravitational field will be given in coming paper.
Liboff L. R. 1990. Kinetic Theory: classical, quantum,and
relativistic descriptions. New Jersey: Prentice-Hall.428.
Landau L D. et al. 1987. The Classical Theory of Fields.
Oxford: Pergmon Press. 29
Huang K. 1963. Statistical Mechanics. John Wiley and Sons,
Inc. 185
Pathria R K.1977. Statistical Mechanics. Bergmon,.